Interestingly, our asymptotic result remains conditionally advantageous even when compared with the first-quantized simulations. There, the best previous approach is the interaction-picture approach [1]} with gate complexity \(\mathcal {O}\left(n^{1/3}\eta ^{8/3}\mathrm {polylog}(n)\right)\) , larger than our new complexity when \(n=\mathcal {O}\left(\eta ^{2-2/(p+1)}\mathrm {polylog}(n)\right)\) . A related approach was described in [1]} based on qubitization, which has a similar performance comparison with our result.It is a subject for future work to compare the concrete resources required by our approach and those required by the first-quantized approach of [3]}. See [tab:resultsummary]Table tab:resultsummary for details.
